Let $(M,g))$ be a Riemannian manifold with the Hodge dual operator $d^{*}$.
Is there a name (and some computation in some reference) for the cohomology of the complex of Harmonic forms with the boundary map $d+d^{*}$.
In the other word: Let $C^{p,q}$ be the space of all harmonic $p+q$ forms on $M$. Assume that $d:C^{p,q}\to C^{p,q+1}$ be the vertical (upward) operator of differentiation and $d^{*}:C^{p,q}\to C^{p,q-1}$ be the horizontal (leftward) operator Hodge dual.
Let $T_{n}=\oplus_{q-p=n} C^{p,q}$. Then the Dirac operator $d+d^{*}:T_{n}\to T_{n+1}$ deefine a complex. Its cohomology is the subject of this question.